Phase-Optimized Constant Envelope Transmission (POCET) Method, Apparatus And System

ABSTRACT

An apparatus and method for generating a composite signal includes electronics configured to modulate a carrier utilizing a finite set of composite signal phases to generate a composite signal, the finite set of composite signal phases being determined through an optimization process that minimizes a constant envelope for the phase modulated carrier, subject to constraints on desired signal power levels of the signals to be combined and either zero or one or more relative phase relationships between the signals. The apparatus and method can be extended to generating an optimized composite signal of different frequencies.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.14/325,159, entitled “Phase-Optimized Constant Envelope Transmission(POCET) Method, Apparatus And System”, filed on Jul. 7, 2014 (now U.S.Pat. No. ______, issued on ______), which is a continuation of U.S.patent application Ser. No. 12/547,383, entitled “Phase-OptimizedConstant Envelope Transmission (POCET) Method, Apparatus And System”,filed on Aug. 25, 2009 (now U.S. Pat. No. 8,774,315, issued on Jul. 8,2014), all of which are hereby incorporated by reference.

STATEMENT OF GOVERNMENT INTEREST

The invention was made with Government support under contract No.FA8802-04-C-0001 by the Department of the Air Force. The Government hascertain rights in the invention.

TECHNICAL FIELD

The invention relates generally to constant envelope transmissions and,in particular, to a Phase-Optimized Constant Envelope Transmission(POCET) technique for generating a composite signal through phasemodulation of a finite set of composite signal phases determined throughoptimization (minimization) of an objective function of the compositesignal envelope, subject to a plurality of intra-signal constraints forthe component signals and either zero or one or more inter-signalconstraints between component signals of the composite signal.

BACKGROUND ART

In prior art constant envelope transmission methods, such as Interplexmodulation, Coherent Adaptive Subcarrier Modulation (CASM),majority-vote modulation, or inter-vote modulation, the best-case powerefficiency is dependent on the relative phase relationships betweencomponent signals and equality or disparity in component signal powerlevels. As discussed below, it would be helpful to be able to improvethe efficiency of constant envelope transmission methods.

In mathematics, the general field of optimization, or mathematicalprogramming, refers to the study of problems in which one seeks tominimize (or maximize) a function known in the art as “an objectivefunction,” sometimes called a “cost function,” f(x), subject to a set ofinequality and/or equality constraint equations, which can be expressedas

g _(j)(x)≦0(Inequality Constraints)

h _(j)(x)=0(Equality Constaints)

for the jth constraint of M total constraint equations. S. S. Rao,“Optimization theory and Applications,” December, 1977.

There are a large number of sub-disciplines in mathematical programmingincluding linear programming, nonlinear programming, quadraticprogramming, convex programming, integer programming, etc., depending onthe form of the objective function. Mordecai Avriel (2003), NonlinearProgramming: Analysis and Methods, Dover Publishing (ISBN0-486-43227-0). Other subdisciplines known in the art describetechniques that optimize the objective function over time, such asdynamic programming or optimal control theory. Techniques within thesefields have not previously been applied to the generation of a constantenvelope modulation. For example, nonlinear programming is a field ofoptimization techniques in which the objective function or theconstraints or both contain nonlinear parts. One example of anoptimization method from nonlinear programming is known as the penaltyfunction method. In this method, an unconstrained minimization (i.e., inwhich the constraint is introduced indirectly) is used to solve aconstrained minimization problem (i.e., where the constraint isintroduced directly) by introducing a penalty factor, μ_(k)>0, and apenalty function p(x)>0 such that a penalized objective function isminimized iteratively (e.g., for penalty factor μ_(k)=1 in the firstiteration, 5 in the 2^(nd) iteration, 10 in the 3^(rd) iteration) sothat the “penalty” for violating the constraint grows after eachiteration. By the end of the process, the minimization is conducted witha large penalty if the constraint is violated; consequently, the resultapproaches the true minimum for introducing the constraints directly.David G. Luenberger (1973), Introduction to Linear and NonlinearProgramming, Addison-Wesley Publishing Company. The form of theobjective function, including the penalty functions and penalty factors,is given below.

${F(x)} = {{f(x)} + {\mu_{k}{\sum\limits_{j = 1}^{m}{G_{j}\left\lbrack {g_{j}(x)} \right\rbrack}}}}$

G_(j)(x) the penalty function and is some function of the j'thconstraintμ_(k) is the penalty parameter corresponding to the k^(th) iterationIt should be noted that there are many variations of the penaltyfunction including absolute value penalty functions, quadratic losspenalty functions, etc. Just as there is no single equation thatdescribes all of the sub-disciplines and methods in the theory ofmathematical programming (optimization), there is also no single penaltyfunction. These are all variations of solving the problem of minimizingan objective function subject to constraints.

In order to maximize efficiency of a nonlinear high-power amplifier, itis preferable to operate such amplifiers at or near saturation of theirnonlinear region. Unfortunately, such operation leads to amplitudemodulation to amplitude modulation (AM/AM) and amplitude modulation tophase modulation (AM/PM) distortions when the amplitude or envelope ofthe composite signal is not constant.

To reduce such distortions in space and ground-based communication andnavigation transmission systems employing nonlinear amplifiers, twoapproaches have been undertaken. The first approach is to operate theamplifier in a linear region through output power backoff. This approachmitigates the effect of AM/AM and AM/PM distortions due to operation inthe nonlinear region but is undesirable because it results in a loss ofnonlinear amplifier efficiency as a result of not operating theamplifier at its maximum output power level (i.e., at saturation).

The second approach is to operate the amplifier in saturation whileminimizing the variation in amplitude of the composite signal, or AMbecause such variation produces undesirable AM/AM and AM/PM distortionswhen the signal is passed through a nonlinear amplifier (as is the casein a typical space-based communication or navigation system). Severalapproaches have been previously proposed to maintain a constant envelopeof the composite signal.

These approaches have been applied to modulate signals in GlobalNavigation Satellite Systems (GNSS) including the Global PositioningSystem (GPS), and the European Galileo navigation satellite system.These approaches maintain a constant carrier envelope according to aspecific signal multiplexing (combining) technique. The approachesinclude: Coherent Adaptive Subcarrier Modulation (CASM), U.S. Pat. No.6,430,213 to Dafesh; Quadrature Product Subcarrier Modulation (QPSM),U.S. Pat. No. 7,120,198 to Dafesh et al.; Interplex modulation, U.S.Pat. No. 6,335,951 to Cangiani et al.; weighted majority vote, U.S. Pat.No. 7,035,245 to Orr et al., or hybrids such as the inter-vote techniqueU.S. Pat. No. 7,154,962 to Cangiani et al. See also: J. Spilker et al.,“Code Multiplexing via Majority Logic for GPS Modernization” Proceedingsof the Institute of Navigation (ION) GPS-98, Sep. 15-18, 1998.; P. A.Dafesh, T. M. Nguyen and S. Lazar, “Coherent Adaptive SubcarrierModulation (CASM) for GPS Modernization,” Proceedings of the IONNational Technical Meeting, January, 1999; and P. A. Dafesh, “QuadratureProduct Subcarrier Modulation,” Proceedings of the IEEE AerospaceConference, March 1999.

For these previous approaches, their efficiency is critically dependenton maintaining certain carrier phases and there is little flexibility tooptimize the phase between different signal components, while minimizingthe output power (producing a best-case power efficiency) required totransmit all of the signal components and pre-specified power levels.Maintaining phase relationships between signals is a desirable featureto meet legacy system requirements, such as the 90 degree phaserelationship between P(Y) and C/A codes in the GPS system or to minimizeinterference between signals that overlap in spectrum.

In the case of GPS or GNSS systems in general, the number of signalsthat must be simultaneously transmitted in the future has increased from2 to at least 5. This has resulted in a need to efficiently transmitthese signals, the preferred method of which has been to employ sometype of combining method that maintains a constant envelope compositesignal amplitude (to eliminate AM-to-AM and AM-to-PM distortions), asdescribed above. This combining approach should provide a compositesignal without deleteriously impacting the power efficiency, defined bythe following equation as the ratio of the sum of the component signalpowers divided by the power of the composite signal:

$\eta = {\left( \frac{\sum\limits_{n = 1}^{N}{{corr}_{n}}^{2}}{A^{2}} \right).}$

Here P_(dn)=|corr_(n)|² is the desired value of the n'th componentsignal power, as measured by a correlation receiver matched to thecomponent signal and P_(T)=A² is the total power of the compositetransmitted signal, where A is the envelope (amplitude) of the compositesignal. Corr_(n) is the expected complex correlation level for the n'thcomponent signal of N signals in the composite signal.

Additional requirements, such as the desire to maintain certain signalphase relationships, allow for adaptive signal power levels; and thepossibility of increasing the number of signals in the future has led toincreased signal losses and potentially severe self interference effectsresulting from inter-modulation products inherent in these prior-artcombining (modulation) methods. The methods also become increasinglydifficult to optimize as the number of signals increases and are notwell suited to combining other than BPSK signals. For example, thedesire to broadcast composite signal envelope including Code DivisionMultiple Access (CDMA) waveforms (composed of multiple orthogonalpseudorandom noise code signals) and Orthogonal Frequency DivisionMultiplexing (OFDM) waveforms (composed of multiple orthogonal componentsignals at different frequencies) using nonlinear high power amplifiersoperating near saturation makes it desirable to develop an efficientmodulation method that may be applied to the optimization of a widerange of signal types, signal levels, phase relationships, and number ofsignals. Such methods may be used to efficiently transmit signals forapplication to both terrestrial wireless and space-based communicationand navigation systems.

SUMMARY OF THE INVENTION

Embodiments described herein relate to a Phase-Optimized ConstantEnvelope Transmission (POCET) modulation methodology that optimizes thecombining of several signal components, subject to a plurality ofintra-signal constraints for the component signals, such as componentsignal power level, and either zero or one or more inter-signalconstraints between component signals, such as the phase relationshipsbetween any two component signals. In an example embodiment, thereceiver for each component of the signal is assumed to correlate withits corresponding pseudorandom noise (PRN) spreading code, withoutknowledge of the other codes. This is possible because the componentsignals being combined are orthogonal in nature (or approximatelyorthogonal). In another example embodiment, different PRN codes areseparated in frequency (i.e., on different carriers) such that thecomposite signal maintains a constant envelope. In alternativeembodiments, other means of orthogonality may be used to separatecomponent signals, such as matched filtering of orthogonal frequencycomponents. This embodiment would apply, for example, to generation ofOFDM in a constant-envelope manner. Many variations of this approach mayalso be possible, including detection of power in individual signalsthrough separation in time, phase, and polarization, as well asapproximately orthogonal patterns in these quantities, as is the case infrequency-hopped spread spectrum signaling. Such component signalseparations allow one to mathematically formulate different powerconstraint equations that relate the complex voltage,(V₁+jV_(Q))_(n)=Re{Corr_(n)}+j Im{Corr_(n)} (or power=|V_(I)+jV_(Q)|_(n)²) of a given signal component to the envelope and carrier phase of thecomposite signal.

In an example CDMA satellite transmitter implementation, a fixed tableof transmitted phase values, one for each combination of bits of binaryPRN spreading codes, is computed. By way of example, the carrier isphase modulated by a complex rotation obtained by table lookup as afunction of the code bits. The table of phase values is obtained throughthe above mentioned optimization scheme to minimize theconstant-envelope transmit power subject to intra-signal constraints ofthe specified component signal power levels. Alternative embodiments mayalso constrain the relative signal phase relationships to provide one ormore inter-signal constraints. For example, the phase difference betweenone or more signals may be constrained to 90 degrees.

The techniques described herein are applicable to efficiently combiningboth spread and unspread signals and can be used to generate a compositesignal having a best-case power efficiency for any arbitrary set ofdesired component signal power levels, independent of the equality ordisparity in component signal power levels. The techniques describedherein produce a constant envelope modulation which is desirable fortransmission of signals through a nonlinear amplifier, such as thoseused in space-based communication systems, in order to avoid deleteriouseffects of amplitude modulation to amplitude modulation (AM/AM) andamplitude modulation to phase modulation (AM/PM) distortions when thecomposite signal is passed through an amplifier operating in itsnonlinear region.

In an example embodiment, an apparatus for generating a composite signalincludes electronics configured to modulate a carrier utilizing a finiteset of composite signal phases to generate a composite signal, thefinite set of composite signal phases being determined through anoptimization process that minimizes a constant envelope for the phasemodulated carrier, subject to intra-signal constraints for the componentsignals (e.g., on desired signal power levels of the signals to becombined). In various embodiments, the optimization process is furthersubject to one or more inter-signal constraints between the componentsignals (e.g., defining the relative phase relationships between thecomponent signals).

In an example embodiment, an apparatus for generating a composite signalfrom a plurality of component signals includes electronics configured tomodulate a carrier utilizing a finite set of composite signal phases togenerate a composite signal, the finite set of composite signal phasesbeing determined through an optimization process that minimizes aconstant envelope of a phase modulated carrier, subject to a pluralityof constraints on desired signal power levels for the component signals.In an example embodiment, the optimization process is further subject toone or more constraints on phase relationships between the componentsignals.

In an example embodiment, a method for generating a composite signalfrom a plurality of component signals includes the step of modulating acarrier utilizing a finite set of composite signal phases to generate acomposite signal, the finite set of composite signal phases beingdetermined through an optimization process that minimizes a constantenvelope of a phase modulated carrier, subject to a plurality ofconstraints on desired signal power levels for the component signals,and either zero or one or more constraints on phase relationshipsbetween the component signals.

A generalized embodiment for the case of PRN spreading signals is thecase where the spreading code is an M-ary modulation with M≧2. Forexample, for QPSK spread spectrum, M=4, for BPSK spread spectrum, M=2.In this case, a method for forming a composite signal from M-arymodulated component signals includes the steps of: enumerating M^(N)phase states for combining N signals with M possible signal phases;formulating N intra-signal constraint equations, each as a function ofthe M^(N) phase states, to provide intra-signal constraints on thesignals; formulating K<=N(N−1)/2 inter-signal constraint equations,where K=0 or a positive integer, each as a function of the M^(N) phasestates, to provide inter-signal constraints between the signals;performing an optimization process to minimize a constant envelope for aphase modulated carrier, subject to the intra-signal constraints on thesignals and the inter-signal constraints between the signals; andmodulating an RF carrier using phase states determined through theoptimization process to produce a composite signal. It should be notedthat K=0 is the special case where there are no inter-signalconstraints. By optimizing the composite signal power, the amplitude(envelope) of the composite signal is also optimized.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an example embodiment of a Phase-Optimized ConstantEnvelope Transmission (POCET) system;

FIG. 2 illustrates an example embodiment of an optimized phase table forfour signals;

FIG. 3 shows a 3 signal, BPSK example of possible phase states;

FIG. 4 is a flow diagram of an example method of forming a compositesignal from N component signals;

FIG. 5 is a plot that shows combining efficiency as a function of thephase difference between the L1C carrier and the GLONASS FDMA carrier;and

FIG. 6 is a plot that shows the power spectral density forconstant-envelope combining of the L1C carrier and the GLONASS FDMAcarrier.

DISCLOSURE OF INVENTION

The invention described herein involves a methodology of multiplexingseveral binary signals using a number of intra-signal (e.g., signalpower) and either zero or one or more inter-signal (e.g., phasedifference between any two signals) constraints to produce aconstant-envelope composite signal with optimized signal phaserelationships and minimum required composite signal power. This approachis called the Phase-Optimized Constant Envelope Transmission (POCET)method. The advantage of this scheme is that it minimizes the requiredcomposite signal power to transmit the selected component signals at thespecified intra-signal constraints for each component signal and eitherzero or one or more inter-signal constraints between component signals.The receiver for any selected signal may operate without knowledge ofthe other signals or signal phase relationships. In general, componentsignal power is one of many possible intra-signal constraints. Themethod is applicable to both spread and unspread signals having a finitenumber of phase values, and any number of component signals can becombined at arbitrary specified power levels and with any number ofphase relationships.

Referring to FIG. 1, in an example embodiment, a POCET system 100includes a RF synthesizer 102, a quadrature modulator 104, an optimizedlookup table 106, sin (θ), cos (θ) generator 108, a high power amplifier110, and an antenna 112 configured as shown. The RF synthesizer 102generates complex carrier signals, which are provided to the quadraturemodulator 104. As described herein, outputs of the binary signalgenerators (S1, S2, . . . SN) are inputs to the optimized lookup table106. The sin (θ), cos (θ) generator 108, in turn, uses outputs from theoptimized lookup table 106 to generate the Q(t) and I(t) inputs to thequadrature modulator 104. The quadrature modulator 104 outputs thecomposite signal, s(t), which is provided as an input to the high poweramplifier 110. An alternative implementation directly phase modulatesthe carrier by the phase θ. For example, in a digital implementation, adigital direct synthesizer (DDS) is configured to directly phasemodulate the carrier by the stored phase value as an implementationalternative to performing the mathematically equivalent QAM modulationshown in FIG. 1.

In an example embodiment, the POCET modulator is configured such thatfour Binary Phase Shift Keying (BPSK) spread signals are combined. ThePOCET modulator can also be extended to a larger number of signals. Forexample, consider the combining of GPS civilian coarse-acquisition(C/A), Modernized L1 civilian signal data and pilot codes signals, L1Cdand L1Cp respectively, and the military precision, P(Y) and Military (M)code signals. These signals may be combined using the herein describedPOCET invention to combine the 5 binary PRN code signals L1Cd, L1Cp,C/A, P(Y) and M code. POCET may be compared to the prior-art Intervoteapproach. In the Intervote approach, assume that (L1Cd,C/A,L1Cp) aremajority voted with M and P(Y) in phase quadrature according to themethods of U.S. Pat. No. 7,154,962. The Intervote technique is known tobe an efficient means of combining 5 signals. Using the POCET modulationmethod, an efficiency improvement (reduction of total power) of roughly10% (0.4 dB) is achieved as compared to the Intervote technique.Moreover, for the case of three signals, POCET is found to provide thesame efficiency of three-signal Interplex, which is known to be the mostefficient technique to combine three signals, but without the ability tooptimize the phase relationships between signals. This is important, forexample, in the case of the legacy GPS P(Y)-code and C/A code rangingsignals that must retain a 90 degree phase relationship.

The POCET techniques described herein involve multiplexing severaldigital signals and provide an optimized combination of any finitenumber of component signals with desired intra-signal constraints (e.g.desired power levels), for a finite number of phase states, subject tozero or one or more desired inter-signal constraints (e.g., specifiedphase differences between selected pairs of signals), while maintaininga constant envelope (composite amplitude) and minimizing the envelope ofthe composite signal and thereby maximizing the transmitter efficiency.In the above mentioned example embodiment (combining 5 bi-phasespreading signals), a fixed storage table of phase values is precomputedto phase modulate the carrier as a deterministic function of the bits ofthe baseband signal components (e.g., 2^(N) phase values with N binarysignals; phase changes when a chip switches polarity). Thus, in anexample embodiment, a result of POCET combining is a deterministic tableof phase values.

In the present example embodiment, binary spreading codes are assumed.

Assuming all N combinations are equally likely, which is valid forapproximately random PRN spreading codes, the average complexcorrelation for signal n, by definition the quantitative measure ofperformance of the correlation receiver for binary signaling, is:

${{Corr}_{n} = {\frac{A}{2^{N}}{\sum\limits_{k = 0}^{2^{N} - 1}{\left\lbrack {1 - {2{b_{n}(k)}}} \right\rbrack {\exp \left( {j\; \theta_{k}} \right)}}}}},$

where A is the constant envelope, θ_(k) is the pre-computed k^(th) phasevalue stored in the table and b_(n)(k) denotes the k-th bit of thebinary representation of N signals being combined. ([1-2 b_(n)(k)]converts b_(n)(k) from a 0/1 representation to +/−1 non-return-to-zerorepresentation). The data bit values have been excluded from the aboveexpression because correlation is performed over a time period which iscoherent with the data bit and therefore only affects the polarity ofthe complex correlation, but not its magnitude (which is used todetermine the power constraint equations).

Note that the {θ_(k)} phase values (k=1, . . . , 2N−1) are determined asa result of the optimization process and take on different values intime depending upon the pseudorandom combination of bits b_(n)(k) forn=1, . . . , N. The implementation of the satellite transmitter causesthe carrier to be phase modulated by rotating the carrier phaseaccording to e^(θ) ^(k) or, equivalently, by quadrature amplitudemodulation of the in-phase and quadrature form, cos (θ_(k))+j sin(θ_(k)), yielding a transmitted signal of s(t) given by:

$\begin{matrix}{{s(t)} = {{A\; {\cos \left( {{2\; \pi \; f_{c}t} + {\theta_{k}(t)}} \right)}} = {A\; {Re}\left\{ {^{j\; {\theta_{k}{(t)}}}^{j\; 2\; \pi \; f_{c}t}} \right\}}}} \\{= {A\; {Re}\left\{ {\left\lbrack {{\cos \; \left( {\theta_{k}(t)} \right)} + {j\; {\sin \left( {\theta_{k}(t)} \right)}}} \right\rbrack ^{j\; 2\; \pi \; f_{c}t}} \right\}}} \\{= {{A\; {\cos \left( {\theta_{k}(t)} \right)}{\cos \left( {2\; \pi \; f_{c}t} \right)}} - {A\; {\sin \left( {\theta_{k}(t)} \right)}{\sin \left( {2\; \pi \; f_{c}t} \right)}}}}\end{matrix}$

where the phase values, θ_(k)=θ_(k)(t), are extracted from the lookuptable as a function of the spreading code bits being combined at anygiven time t, and A is the envelope of the composite signal, s(t), alsocalled its amplitude. The amplitude A is >0.

The lookup table of phase values is determined from an optimizationalgorithm to minimize the constant envelope A, while maintaining theconstraints of the specified power levels and phase relationships of theaverage correlations. FIG. 2 illustrates an example of an optimizedphase table for four signals, which enumerates the possible values ofb_(n)(k). In this example, N=4, and the columns are numbered from leftto right, n=1, 2, 3, 4. The transmitted phase angles are listed in therightmost column from k=0, 1, 2 . . . , 15 and corresponding to eachpossible combination of the 4 bit values from the binary PRN codesignals: C/A, P(Y), L1Cp, and L1Cd. A new phase table is produced foreach set of inter-signal and intra-signal constraints.

In an example embodiment, an algorithm that performs the constrainedoptimization is based on an unconstrained optimization search of theobjective function that incorporates the constraints according to thepenalty function technique, where the penalty function is zero when theconstraints are satisfied and positive otherwise. A quasi-Newton searchalgorithm can be used to minimize the objective function of severalvariables without requiring explicit computation of the gradient of thefunction. Other optimization approaches can also be used. In this case,the variables are the phase angles of the look-up table and the envelopeA to be minimized. The phase angle values are significant modulo-360°but are unconstrained for the search. A common problem with theminimization search is that the algorithm may converge to a localminimum rather than the desired global minimum. An approach tocircumvent this difficulty is to perform a multiplicity of searchesstarting from different starting points, where each starting pointinitializes the phase values to random values in the range 0 to 360°.With N binary codes, the number of independent variables for theoptimization search is 2^(N-1)−1, because the phase values occur insymmetrical pairs differing by 180° and one phase value can bearbitrarily set to zero, as is evident from table 2. The total number ofphase values in the lookup table is 2^(N). In the example above, theoptimization is performed over 7 independent phase angles.

An example multiplexing scheme to combine GPS navigation signalsprovides an efficient method of optimally combining three or moredigital signals into a composite signal of constant envelope, withminimum total power, and with arbitrary power and phase constraints,such as requiring the GPS C/A code signal to be in phase quadrature (90degrees out of phase) with the P(Y) code. The features of this methodare not specific to its application to GPS or a fixed number of signals.

In an example embodiment, a constrained optimization (nonlinearprogramming problem) is performed as set forth below:

Minimize A as a function of θ_(k) for k=0, . . . , 2^(N-1)−1, subject tothe constraints

${P_{dn} = {{{{corr}_{n}(\theta)}}^{2} = {\frac{A^{2}}{2^{N}}{{\sum\limits_{k = 0}^{2^{N} - 1}{\left\lbrack {1 - {2{b_{n}(k)}}} \right\rbrack {\exp \left( {j\; \theta_{k}} \right)}}}}}}},$

for desired power levels, and

Im{e^(−jΔφ) ^(nl) corr_(n)(θ)corr_(l)(θ)^(*)}=0

Re{e^(−jΔφ) ^(nl) corr_(n)(θ)corr_(l)(θ)^(*)}>0

are equations defining an inter-signal phase constraint betweencomponent signals, where θ=

θ₀, . . . , θ₂ _(N-1)

is a vector containing all of the possible composite signal phases, asexemplified transmitted phase angle values in FIG. 2.

Where:

A is the constant envelope of phase-modulated carrier;

θ_(k) is phase value for k^(th) combination of binary chips at timet_(k);

corr_(n)(θ) is the complex correlation of the composite signal to then^(th) replica of component signal n;

corr_(l)(θ) is the complex correlation of the composite signal to thel^(th) replica of component signal l; and

Δφ_(nl) is the phase difference inter-signal constraint betweencomponent signals k and l.

t_(k+1)−t_(k)=Δt_(k) is the duration of the shortest PRN chip of the NPRN codes to be combined, where the conventional definition of a chip isused corresponding to the smallest interval of time over which thesignal transmitted component signal is constant. It should be noted thatan alternative embodiment minimizes A² instead of A.

In an example embodiment, it is desirable to combine 4 GPS PRN codesignals including C/A code, P(Y) code, L1Cp code and the L1Cd code. Forthis case, the intra-signal power constraints are specified by:

C/A=0 dB;

P(Y)=−3 dB;

L1C_(P)=0.25 dB; and

L1C_(d)=−4.5 dB.

And the specified phase constraints are:

C/A to P(Y)=90°;

C/A to L1C_(p)=90°; and

C/A to L1Cd=90°.

In the example embodiments, methods of combining three or more componentsignals have applicability to global navigational satellite systems(GNSS) and to other spread spectrum signaling in which more than twosignals are combined with inter-signal phase constraints in aconstant-envelope composite signal having minimum power. In exampleembodiments, the methods of signal combining described herein areapplicable to modulation of multiple wireless communication signals in aconstant envelope composite signal including Orthogonal FrequencyDivision Multiplexing (OFDM) or Code Division Multiple Access (CDMA)systems and signaling standards, or other standards that would beapparent to one skilled in the art.

In an alternative embodiment, the average component signal correlationfor a composite signal comprising N, M-ary spreading signals (M^(N)phase states) is determined as follows:

${Corr}_{n} = {\frac{A}{M^{N}}{\sum\limits_{k = 0}^{M^{N} - 1}{\left\lbrack {{C_{In}(k)} - {j\; {C_{Qn}(k)}}} \right\rbrack {\exp \left( {j\; \theta_{k}} \right)}}}}$

For component signals employing binary signaling, M=2. For signalsemploying QPSK spreading codes, M=4. Here C_(I)(k)−jC_(Q)(k) is thecomplex representation of an M-ary spreading code. The data bit valueshave been excluded from the above expression because correlation isperformed over a time-period, which is coherent with the data bit andtherefore only affects the polarity of the complex correlation, but notits magnitude (which is used to determine the intra-signal powerconstraint equations).

The general case applies to any M-ary digital modulation, e.g., QPSK(M=4), 8-PSK (M=8), etc.

In an example embodiment, 5 spread spectrum signals are combined into acomposite CDMA signal; however, the general method applies to acombination of forms of multiplexed signals (e.g., Frequency DivisionMultiple Access or FDMA) or any other plurality of component signals. Inthis 5 signal binary example, the average component signal correlationfor a composite signal is determined as follows:

${Corr}_{n} = {\frac{A}{2^{N}}{\sum\limits_{k = 0}^{2^{N} - 1}{C_{I_{n}{(k)}}{\exp \left( {j\; \theta_{k}} \right)}}}}$C_(I_(n)(k)) = 1 − 2b_(n)(k) = ±1${Corr}_{n} = {{\frac{A}{32}{\sum\limits_{k = 0}^{31}{\left\lbrack {1 - {2{b_{n}(k)}}} \right\rbrack {\exp \left( {j\; \theta_{k}} \right)}}}} = {\frac{A}{16}{\sum\limits_{k = 0}^{15}{\left\lbrack {1 - {2{b_{n}(k)}}} \right\rbrack {\exp \left( {j\; \theta_{k}} \right)}}}}}$

In this example, the series is reduced as shown with symmetry being usedto reduce the number of independent values of θ_(k) from 32 to 16, andone of these phases may be set to 0 degrees as an arbitrary referencevalue.

FIG. 3 shows a 3 signal, BPSK example of possible phase states. The 3signal table is shown for the sake of simplicity and convenience. For a5 signal example (N=5), the table would include two additional columnsand a total of 32 (2^(N)) rows.

Further with respect to the optimization (e.g., minimization) of anobjective function, in an example embodiment, an objective function,f(x), is subject to a set of inequality and/or equality constraintequations, which can be expressed as

g _(j)(x)≦0(Inequality Constraints)

h _(j)(x)=0(Equality Constraints)

wherein j=1, 2, . . . N.

Various optimization sub-disciplines including nonlinear programming canbe used where the objective function is nonlinear. In an exampleembodiment, a penalty function method is employed as the optimizationapproach. By way of example,

${F(x)} = {{f(x)} + {\mu_{k}{\sum\limits_{j = 1}^{N}{G_{j}\left\lbrack {g_{j}(x)} \right\rbrack}}}}$

G_(j)(x) the penalty functon and is some function of the j'th constraint

μ_(k) is the penalty parameter corresponding to the k^(th) iteration

$\mspace{166mu} {\left. {{Power}\mspace{14mu} {to}\mspace{14mu} {be}\mspace{14mu} {minimized}}\mspace{115mu}\swarrow {F(\theta)} \right. = {A^{2} + {\mu_{a}{\sum\limits_{n = 1}^{m}{\overset{{Received}\mspace{14mu} {power}\mspace{14mu} {constraints}}{\left( {{{{corr}_{n}(\theta)}} - {corrd}_{n}} \right)^{2} +}\mu_{b}{\sum\limits_{n = 1}^{N}{\sum\limits_{l = {n + 1}}^{N}\overset{{Phase}\mspace{14mu} {constraints}}{{Im}\left\{ {^{{- j}\; \Delta \; \varphi_{nl}}{{corr}_{n}(\theta)}{{corr}_{l}(\theta)}^{*}} \right\}^{2}}}}}}}}}$     Δ φ_(nl) = Desired  inter-signal  phase  constraints  between  signals     θ = (θ₁, θ₂, θ₃, …  θ_(M))

and Corrd_(n) is the desired intra-signal power level for each componentsignal and is determined from the desired component signal power level.Other optimizations and penalty functions may also be employed.

Example applications of the methodologies described herein include, butare not limited to, the modulation of N different spreading signals suchas is the case for the modernized GPS system, the European Galileonavigation, or other future foreign navigation systems such as China'sCompass system. The methodologies described herein are generallyapplicable to the generation of a constant-envelope composite signal forany spread-spectrum (CDMA) system including wireless CDMA signalingstandards employed in 2.5G and 3G wireless standards. The methodologiesdescribed herein can also be applied to OFDM, a technology expected tobe used in 4G and LTE wireless or similar systems (cell phones, wirelessrouters, etc.), as well as transmissions of these signals from space(through a nonlinear amplifier).

Referring to FIG. 4, in an example embodiment, a method 400 of forming acomposite signal from N component signals is represented in the form ofa flow diagram. At 402, all M^(N) possible phase states for combining Nsignals each having M possible signal phases (e.g., binary M=2) areenumerated. At 404, N intra-signal constraint equations are formed fromintra-signal (e.g., desired power) constraints, each as a function ofthe M^(N) possible phase states. At 406, K<=N(N−1)/2 inter-signalconstraint equations, where K=0 or a positive integer, are formed frominter-signal (e.g. phase difference) constraints between signals, eachas a function of the M^(N) possible phase states. Thus, either zero orone or more inter-signal constraints are formed. At 408, the constantsignal envelope is optimized (minimized) as a function of the M^(N)carrier phases, subject to the intra-signal constraints on the signalsand the inter-signal constraints between signals. At 410, an RF carrieris modulated using the phase states determined through the optimizationprocess. Alternative inter-signal and intra-signal constraints may alsobe used.

As previously described, POCET enables combining of several binarysignals into a constant envelope while maintaining desired intra-signaland inter-signal constraints. To combine several binary signals, POCET,as previously described, stores a pre-computed phase table that is afunction of the values of the binary chips of the signals to becombined. For example, with four binary signals, the phase table has2⁴=16 entries, each entry specifying the transmitted phase correspondingto one of the possible set of values of the four binary chips. Presumingrandom chip values, a computational optimization technique minimizes therequired constant-envelope power (=A²) while maintaining the specifiedintra-signal and intra-signal constraints. In an example embodiment, theintra-signal constraints are the desired component signal powers andintra-signal constraints are the desired phase difference constraintsbetween signals (relative phase constraints). As previously developed,POCET assumed that these constraints do not vary in time. In ageneralization of the POCET scheme, the constraints may also be timevarying. For the example of inter-signal phase constraints betweensignals, this generalization may be used to combine component PRNspreading code signals at different frequencies to produce aconstant-envelope signal. In such a variation, additional optimizations(or phase tables) are generated corresponding to the number of phasesteps used to represent a frequency difference between componentsignals. The following describes a technique to combine signals atdifferent carrier frequencies onto a constant-envelope carrier suitablefor efficient high-power transmission from a satellite.

The capability of POCET to maintain one or more phase constraints isnecessary to enable the extension of this technique described below.While the following descriptive example refers to combining chips ofbinary PN spread-spectrum signals typically used for navigation andcommunication, it should be understood that the technique can begeneralized to higher-order modulations.

The process of combining two or more signals of different carrierfrequencies into a constant envelope takes into consideration the factthat the phase relation between the signals is continually changing at arate equal to the difference of the carrier frequencies. Therefore, thephase difference constraint, Δφ_(nl), takes on V different values,Δφ_(nl)=Δφ_(n1), Δφ_(n2), . . . Δφ_(nP), where P is the number of phasesteps used to determine the carrier frequency difference betweencomponent signals and one optimized POCET composite carrier phase tableis needed for each phase step.

In an example embodiment, the POCET technique is extended to thissignal-combining problem by quantizing the continuous phase rotationinto finite phase steps and then performing a POCET optimization of thecombining for the fixed phase relation between the signals at each phasestep. Because this requires multiple POCET solutions for the differentphase relations, in an example embodiment, relatively coarse phase stepscan be used. For example, quantizing the phase rotation to steps of 10°ensures a negligible correlation loss, calculated to be 0.01 dBpresuming a uniform distribution of phase error with respect to thecontinuous phase rotation. Taking the example of two carrier frequenciesto be combined, using a phase step of 10°, 18 phase tables are computedand stored for one cycle of phase rotation, noting that when the phaserotation increases by 180°, the binary chip values are complemented. Forimplementation convenience, in an example embodiment, the number ofphase tables can be a power of 2, as discussed below.

An example of combining signals at different frequencies for a satelliteimplementation is now described. More specifically, in this example, thesatellite implementation is configured to simultaneously transmit boththe L1C navigation signal and the Global Navigation Satellite System(GLONASS) FDMA signal via a constant-envelope combined signal, whichmaximizes the efficiency of the high-power transmitter. The carrierfrequency of the L1C signal is 1575.42 MHz. The chip rate of the L1Cbinary modulation is 1.023 Mchips/sec with BOC(1,1) chip modulations onthe L1Cp signal and the L1Cd signal. The L1Cp signal also has a timemultiplex of BOC(6,1). The carrier frequency of GLONASS FDMA is one of14 values spaced by 0.5625 MHz in a band centered at about 1602 MHz. Thehighest GLONASS FDMA carrier frequency is 1605.375 MHz. The GLONASS FDMAsignal combines two binary modulations, a civil signal with BPSK chipmodulation at 0.511 Mchips/sec and a military signal with BPSK chipmodulation at 5.11 Mchips/sec.

In this example, there are four signals to be combined. The L1Cd andL1Cp signals can be either at a relative phase shift of 0 or 90°. Therelative phase shift between the civil and military GLONASS FDMA signalsis assumed to be 90°. The GLONASS FDMA carrier must be represented witha minimum of two samples per cycle of phase rotation. A phase-steppingrate of 98.208 MHz=1.023 MHz×6×16 gives 3.28 samples per cycle and is aconvenient multiple of the switching rate of the BOC(6,1) chipmodulation. This sampling rate can be selected to combine the signals.In this example, the specified powers are of −158.25 dBW for L1Cp, −163dBW for L2Cd, −158.5 dBW for GLONASS FDMA civil, and −161.5 dBW forGLONASS FDMA military. In this example, a more efficient combiningresult for POCET is obtained specifying a phase shift of 0° between L1Cpand L1Cd rather than 90°. A POCET optimization was performed for each of18 phase steps from 0 to 170° for the phase rotation of the GLONASS FDMAcarrier relative to the L1C carrier. In the previously described POCEToptimization, the first phase value in the phase table was set to zero;however, this is arbitrary. In this example embodiment, after obtainingthe optimization solution at each phase step, the solution isphase-shifted by adding a constant to all of the phase values stored inthe phase table so that the average correlation of L1Cp is maintained ata phase of 0° as the GLONASS FDMA carrier rotates.

FIG. 5 is a plot that shows combining efficiency as a function of thephase difference between the L1C carrier and the GLONASS FDMA carrier.The symmetry of the plot is due to the fact that the combiningefficiency is the same for negative and positive phase differences.

The signal combining of this example was simulated at the sampling rateof 98.208 MHz. The assumed GLONASS FDMA carrier frequency=1605.375 MHz,which is the maximum offset from the L1 carrier of 1575.42 MHz and,therefore, the most stringent for the digital implementation. Thesimulation incorporated 18 POCET phase tables, one table for eachquantized step of the phase rotation between the two carriers. Assumingrandom binary codes, the average correlation was computed for each ofthe four signals, summing at the sampling rate. The simulation computedthe GLONASS FDMA correlations assuming a continuously-rotating phase ofthe replica carrier in the receiver at each sample rather than thequantized phase shift used for table lookup in the modulator. Theswitching times of the L1C binary modulations were synchronous with thesampling rate. The switching times of the GLONASS FDMA binarymodulations were slightly displaced as necessary to synchronize with thesampling rate. The effect of this quantization of the chip switchingtime was ignored when computing the GLONASS FDMA correlations. The powerspectral density of the combined signal was computed using a FFT of size8192, which gave a frequency bin resolution of 98.208 MHz/8192=11.99KHz. The power from many successive FFT computations was averaged over along simulation run.

Referenced to the constant-envelope transmitted power of the combinedsignal, the simulation produced correlation powers of −5.67 dB for L1Cp,−10.41 dB for L1Cd, −5.93 dB for GLONASS FDMA civil, and −8.94 dB forGLONASS FDMA military. As desired, the phase shift between the L1Cp andthe L1Cd average correlations is almost exactly 0°, and the phase shiftbetween the GLONASS FDMA civil and the GLONASS FDMA military averagecorrelations is almost exactly 90°. Because ideal linear combining ofthe four signals would produce a correlation power of −4.40 dB for L1Cp,the combining loss is 1.27 dB.

FIG. 6 is a plot that shows the power spectral density obtained from thesimulation. The BOC(6,1) chip modulation was not included in thissimulation, although it would be a straightforward addition. Thespectral shape of BOC(1,1) is evident at the L1 carrier. At the GLONASSFDMA carrier, the spectral shape is the sum of BPSK(1) and BPSK(10). Thecombining does produce a low level of spurious components.

In this example, the L1 carrier frequency and the code rate clock todrive the L1C code generators can be synthesized from a frequencystandard at 10.23 MHz. This frequency standard can be divided down to1.023 MHz to drive the L1C code generators, multiplied by 2 to provideBOC(1,1) subcarrier switching, further multiplied by 6 to provideBOC(6,1) subcarrier switching, and additionally multiplied by 8 toprovide the sampling rate of 98.208 MHz. The frequency standard can bedivided down to 1 KHz, multiplied to 0.511 MHz, and further multipliedto 5.11 MHz to produce the code clocks to drive the GLONASS FDMA codegenerators. In an example embodiment, the GLONASS FDMA carrier frequencydoes not physically exist separate from the POCET modulator circuitry,rather its instantaneous phase is digitally represented in a carriernumerically-controlled oscillator (NCO) accumulating at the samplingrate of 98.208 MHz. This approach results in a composite signal thatcontains a digitally generated FDMA carrier modulated by the compositeFDMA spreading signals. In an example embodiment, the number of bits inthe phase accumulator of the NCO must be sufficiently high to obtain therequisite frequency accuracy, for example, 36 bits enables a GLONASSFDMA carrier frequency accuracy of 10^(−10.8). To maintain coherencebetween carrier and code, the GLONASS FDMA code-rate clock can bederived from a NCO accumulating at the sampling rate of 98.208 MHzinstead of using the synthesis scheme described above.

In an example embodiment, the GLONASS FDMA carrier phase in the carrierNCO is used to select the correct POCET phase table. If the number ofPOCET tables is a power of 2, the selection can be done from the MSBs ofthe digital phase representation in the NCO, e.g., 5 MSBs will pick oneof 32 phase tables. In the simulation, the GLONASS FDMA carrier NCO andcode-rate NCO phase were computed by incrementing double-precisionvariables, thus providing 51 bits of accuracy for these frequencies.

The POCET extension to the problem of combining a multiplicity ofsignals at different carrier frequencies into a constant-envelopecarrier is illustrated by studying the afore-described example ofcombining the L1C signal and a GLONASS FDMA signal, which have carrierfrequencies differing by a maximum of about 30 MHz. In this example, thePOCET optimization was performed at a finite number of quantized phaseshifts between the carriers, as an approximation to continuous phaserotation. The combining result for this example had a combining loss of1.27 dB and produced spurious components that were reasonably low. Theimplementation for this example used a sampling rate of 98.208 MHz.Frequency synthesis was from a 10.23 MHz standard to generate the L1Ccarrier and code rates. The GLONASS FDMA carrier and code rates weregenerated by NCOs accumulating at the sampling rate.

Although the present invention has been described in terms of theexample embodiments above, numerous modifications and/or additions tothe above-described embodiments would be readily apparent to one skilledin the art. It is intended that the scope of the present inventionextend to all such modifications and/or additions.

1. An apparatus for generating a constant envelope composite signal, comprising: a modulator configured to phase modulate a carrier signal utilizing a finite set of composite signal phases to combine three or more component signals, each having two or more phases, in order to generate and maintain the constant envelope composite signal; and computation circuitry configured to determine the finite set of composite signal phases through an optimization process that minimizes the power of the composite signal, subject to a plurality of intra-signal constraints for the component signals, said optimization process includes optimization of an objective function of the power of the composite signal.
 2. The apparatus for generating a composite signal of claim 1, wherein the optimization process is further subject to one or more inter-signal constraints between the component signals.
 3. (canceled)
 4. The apparatus for generating the composite signal of claim 2, wherein the inter-signal constraints are selected to minimize interference between signals that overlap in spectrum.
 5. The apparatus for generating the composite signal of claim 1, wherein the component signals are M-ary digital modulations.
 6. The apparatus for generating the composite signal claim 5, wherein the optimization process is further subject to one or more inter-signal constraints between the component signals.
 7. The apparatus for generating the composite signal of claim 4, wherein the component signals are M-ary digital modulations.
 8. The apparatus for generating the composite signal of claim 4, wherein the component signals include Orthogonal Frequency Division Multiplexing (OFDM) component signals.
 9. The apparatus for generating the composite signal of claim 4, wherein the component signals include Code Division Multiple Access (CDMA) component signals.
 10. The apparatus for generating the composite signal of claim 4, wherein the component signals are pseudorandom noise (PRN) spreading codes.
 11. The apparatus for generating the composite signal of claim 10, wherein the PRN spreading codes are at a plurality of different carrier frequencies.
 12. The apparatus for generating the composite signal of claim 4, wherein the component signals include satellite navigation signals.
 13. The apparatus for generating the composite signal of claim 4, wherein the component signals are at a plurality of carrier frequencies. 14-23. (canceled)
 24. The apparatus for generating the composite signal of claim 5, wherein the component signals include Orthogonal Frequency Division Multiplexing (OFDM) component signals.
 25. The apparatus for generating the composite signal of claim 5, wherein the component signals include Code Division Multiple Access (CDMA) component signals.
 26. The apparatus for generating the composite signal of claim 5, wherein the component signals are pseudorandom noise (PRN) spreading codes.
 27. The apparatus for generating the composite signal of claim 26, wherein the PRN spreading codes are at a plurality of different carrier frequencies.
 28. The apparatus for generating the composite signal of claim 5, wherein the component signals include satellite navigation signals.
 29. The apparatus for generating the composite signal of claim 5, wherein the component signals are at a plurality of carrier frequencies. 30-32. (canceled)
 33. A method for generating a constant envelope composite signal, comprising the steps of: phase modulating a carrier signal with a modulator that uses a finite set of composite signal phases to combine three or more component signals, each having two or more phases, in order to generate the constant envelope composite signal; and determining with computation circuitry, the finite set of composite signal phases through an optimization process that minimizes the power of the composite signal, subject to a plurality of intra-signal constraints for the component signals, said optimization process includes optimization of an objective function of the power of the composite signal.
 34. (canceled)
 35. The method for generating the composite signal of claim 33, wherein the optimization process is further subject to one or more inter-signal constraints between the component signals.
 36. The method for generating the composite signal of claim 35, wherein the inter-signal constraints are selected to minimize interference between signals that overlap in spectrum.
 37. The method for generating the composite signal of claim 33, wherein the component signals are M-ary digital modulations. 38-40. (canceled)
 41. The method for forming the composite signal of claim 37, wherein the optimization process is further subject to one or more inter-signal constraints between the component signals.
 42. The method for forming the composite signal of claim 36, wherein the component signals are M-ary digital modulations.
 43. The method for forming the composite signal of claim 36, wherein the component signals include Orthogonal Frequency Division Multiplexing (OFDM) component signals.
 44. The method for forming the composite signal of claim 36, wherein the component signals include Code Division Multiple Access (CDMA) component signals.
 45. The method for generating the composite signal of claim 36, wherein the component signals are pseudorandom noise (PRN) spreading codes.
 46. The method for generating the composite signal of claim 45, wherein the PRN spreading codes are at a plurality of different carrier frequencies.
 47. The method for generating the composite signal of claim 36, wherein the component signals include satellite navigation signals.
 48. The method for forming the composite signal of claim 36, wherein the component signals are at a plurality of carrier frequencies. 49-54. (canceled)
 55. The method for forming the composite signal of claim 37, wherein the component signals include Orthogonal Frequency Division Multiplexing (OFDM) component signals.
 56. The method for forming the composite signal of claim 37, wherein the component signals include Code Division Multiple Access (CDMA) component signals.
 57. The method for forming the composite signal of claim 37, wherein the component signals are pseudorandom noise (PRN) spreading codes.
 58. The method for generating the composite signal of claim 57, wherein the PRN spreading codes are at a plurality of different carrier frequencies.
 59. The method for forming the composite signal of claim 37, wherein the component signals include satellite navigation signals.
 60. The method for forming the composite signal of claim 37, wherein the component signals are at a plurality of carrier frequencies. 